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In mathematics, the main conjecture of Iwasawa theory is a theorem in Iwasawa theory which demonstrates that there is a strong relation between special values of the Riemann zeta function (or more generally Dirichlet L-functions) and the arithmetic of cyclotomic fields. It was first conjectured by Kenkichi Iwasawa^[1] and proved in simple cases by him. The first proof in general (for primes p > 2) was obtained in joint work of Barry Mazur and Andrew Wiles in the early 1980s (Mazur & Wiles 1984). In later work (Wiles 1990), Wiles treated the case p = 2. The statement itself is a generalization of Kummer's criterion on the regularity of p in terms of Bernoulli numbers, as well as the Herbrand–Ribet theorem. It says that the analytic p-adic L-function of Tomio Kubota and Heinrich-Wolfgang Leopoldt defined by interpolating special values of Dirichlet L-functions is essentially the same as the arithmetic p-adic L-function constructed by Iwasawa using the Galois module structure of class groups in towers of cyclotomic fields.

Increasingly broad generalizations and analogues of the original conjecture involving the equality of analytic and arithmetic p-adic L-functions have appeared, all going under the name main conjecture of Iwasawa theory (sometimes with other terms added). Generalizations involve: changing the number fields in question, changing the Galois modules being studied, replacing the tower of cyclotomic fields with more general general towers (or p-adic moduli spaces), or any combination of the above.

Notes

^ (Iwasawa 1964) and (Iwasawa 1969) are typically cited as the articles from which the conjecture originates.

References

Iwasawa, Kenkichi (1964), "On some modules in the theory of cyclotomic fields", Journal of the Mathematical Society of Japan, 16: 42–82, doi:10.2969/jmsj/01610042, MR 0215811
Iwasawa, Kenkichi (1969), "On p-adic L-functions", Annals of Mathematics, 82: 198–205, doi:10.2307/1970817, MR 0269627
Mazur, Barry; Wiles, Andrew (1984), "Class fields of abelian extensions of Q", Inventiones Mathematicae, 76 (2): 179–330, doi:10.1007/BF01388599, MR 0742853
Wiles, Andrew (1990), "The Iwasawa conjecture for totally real fields", Annals of Mathematics, 131 (3): 493–540, doi:10.2307/1971468, MR 1053488

[1] (Iwasawa 1964) and (Iwasawa 1969) are typically cited as the articles from which the conjecture originates.

[1]